Abstract

We show that there exist large algebraic structures (vector spaces, algebras, closed subspaces, etc.) formed entirely (except for 0), on one hand, by singular, nowhere monotonic functions on [0,1] and, on the other hand, by absolutely continuous nowhere monotonic functions. Several tools, of independent interest, related to obtaining special partitions of R\mathbb{R} into uncountable collections will be provided and used. The results obtained in this note are either new or improved versions of already existing ones.

Author information

Contact details are reproduced from the original publication and may be historical.

Luis Bernal-González

Dep. de Análisis Matemático, Facultad de Matemáticas, Inst. de Matemáticas Antonio de Castro Brzezicki, Universidad de Sevilla, Spain

lbernal@us.es

Juan B. Seoane-Sepúlveda

Inst. de Matemática Interdisciplinar, Dep. de Análisis Matemático y Matemática Aplicada, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Spain

jseoane@ucm.es

Keywords

  • Lineability
  • spaceability
  • bounded variation function
  • singular function
  • absolutely continuous function
  • nowhere monotonic function

Mathematics Subject Classification

15A03, 46B87, 26A45, 26B30

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Suggested citation

L. Bernal-González, J. Fernández-Sánchez, J. B. Seoane-Sepúlveda, W. Trutschnig. “On Special Partitions of [0,1] and Lineability within Families of Bounded Variation Functions.” Journal of Convex Analysis 30 (2023), No. 1, 65–80. https://doi.org/10.68381/jca30005

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